From the first relation in eq. (REF ) and the \(3\sigma \) 
range of \(\theta _{13}\)  from Ref. [1]}, we deduce that \(\theta \)  lies in the range \(0.1763\le \theta \le 0.1920\) . Moreover, by using the \(3\sigma \)  ranges of \(\Delta m_{ij}^{2}=m_{i}^{2}-m_{j}^{2}\)  in the case of the IH and the neutrino masses in eq. (REF ) as well
as the constraint on the sum of \(m_i\)  from cosmological
observations \(\sum m_{i}<0.12\)  \(\mathrm {eV}\)  [2]}, we find that \(\theta \)  lies in the range \(0.398\lesssim \theta \lesssim 0.579\)  which
indicates that \(\sin ^{2}\theta _{13}\)  and \(\sin ^{2}\theta _{12}\)  fall far
outside their allowed \(3\sigma \)  experimental range. For this reason, the IH
pattern is excluded in our model.
